THE DEVELOPMENT OF WESTERN TUNING SYSTEMS AND THE CONTEMPORARY USE OF EXTENDED JUST INTONATION

Momilani Ramstrum

A Research Paper Presented to the Dr. David Ward-Steinman

INTRODUCTION

Background and Need for Study

Over the last 2500 years, the tuning systems used in western music have evolved through four major changes; Pythagorean tuning, just intonation, meantone / well temperament, and equal temperament. For the past one hundred years, the predominant tuning system in western music has been equal temperament. Despite the uniform and unrestricted modulatory freedom that accompanied the widespread adoption of equal-temperament tuning, there is a growing number of contemporary musicians who insist on alternative tuning systems to realize their musical goals. In this paper, we will begin by exploring the historical evolution of western tuning systems. Then, we will examine the practices of some modern composers who have turned to extended just intonation, a variation of a Renaissance tuning system, and examine if their compositional objectives could have been met within the realm of equal temperament.

Purpose

There are two purposes for this paper:

1) to explore the development and historical usage of western tuning systems.

2) to observe how extended just intonation is used by some contemporary musicians and determine whether their use of just intonation is an acoustic preference, essential to their compositional process, or if their compositional goals could have been achieved using equal temperament.

Limitations

This bibliographic research will be limited to looking at the historical evolvement of tuning systems and their use in western music. We will the examine the ideologies and compositional practices of Harry Partch and Ben Johnston to see how they use just intonation in their creative work.

Methodology

A review of literature will be undertaken by this author to investigate the origins and common usage of tuning systems in western music. We will look at the acoustic basis of this system, and how and when it developed.

Two contemporary composers that employ just intonation systems in their compositional processes will be observed to note ideologies and motivation for using alternatives to equal temperament. Historical material about each composer will be covered as it pertains to their choice of tuning systems.

Definition of Terms

The following information on just intonation and equal temperament has been compiled from The Oxford Concise Dictionary of Music, Source Readings in Music History, Strobe Tuner Settings for the Historical Scales, and articles by Bill Alves, Jim Campbell and David Canright from the World Wide Web.

Tuning: A tuning is different than a temperament. In a tuning, all intervals are pure and derived from the harmonic series. Tuning produces just intonation. In just tuning, while tuning certain intervals to be pure, this results in other intervals that are not pure. This is the case in Pythagorean tuning where all intervals are constructed by a fifth relation. The resultant pitches are all purely intonated perfect fifths, but, not purely tuned thirds. Just intonation is also known as the 'untempered scale' and is made up of "the pure tones of the unequally divided octave." [1] The intervals in a tuning can all be expressed as a ratio of two integers, e.g. the ratio 4/3 is a pure perfect fourth and 3/2 is a pure perfect fifth. [2] These ratios were first thought to be discovered by Pythagoras as he passed by a smithy shop and heard that the "strikings of mallets upon the anvil were dissonant and consonant." [3] He discovered that the differences were due to a ratio that was proportional to their weights. [4] The ratio of 3:2 produced an interval of the fifth. To generate a scale, you start with a note and produce perfect fifths above it for seven octaves. This will generate the twelve notes of the chromatic scale and end at about the same note as you began (seven octaves up). Instead of being exactly at the same note, you are actually 24 cents or a quarter of a semi-tone off. [5] This difference is called a comma. In equal temperament, to compensate for this comma, each fifth was flatted by 2 cents. "When the resulting twelve notes are all shifted, by octaves, into ascending order, the result is an equal tempered chromatic scale. The spacing between the notes of this scale will have intervals which are all equal. " [6]

Temperament: With temperament, a modification of tuning is needed and radical numbers are used to express some or all of the intervals, [7] e.g. 1.3348 is a perfect fourth and 1.4983 is a perfect fifth. [8] Temperament has been called "controlled mistuning" [9] and is used to compensate for the acoustic phenomena that all perfect fifths and perfect octaves in all keys can not exist at the same time.

Restricted: A tuning or temperament that does not allow complete modulatory freedom because of the presence of 'wolf' intervals that makes certain keys unusable, [10] e.g. just intonation or meantone temperament.

Unrestricted: A tuning or temperament that does allow complete modulatory freedom and can be played in all keys, [11] e.g. well temperament or equal temperament.

Regular: A tuning or temperament where all the intervals are tempered by the same amount, [12] e.g. equal temperament.

Irregular: A tuning or temperament where the intervals are tempered by the different amounts, [13] e.g. just intonation or well temperament.

Closed system: A system in which you end up where you began. For example, in equal temperament at an octave you end up at the same note. All tempered systems create a closed system.

Open system: A system where you will continually have a slight discrepancy between the purely intonated note and its nearest enharmonic neighbor. In all tunings, this is the case. When you ascend by a pure just interval, you will never get exactly to any other pure just interval. The difference is known as a comma. Pythagorean tuning is an open system.

Summary

Tuning systems in western music have changed dramatically over the past 2500 years. We have progressed from the predominance of pure intervals as in Pythagorean and just intonation, to an exclusion of all pure intervals, but the octave, with the prevalence of equal temperament. Some contemporary composers have returned to a form of just intonation as a way

of producing purer tones, to generate pitch sets or to exploit just relationships

over constant fundamentals. [14] In this paper, this author will examine the history of Western tuning systems and explore how some of today's composers are using extended just intonation as a compositional tool.

REVIEW OF LITERATURE

History Of Western Tuning Systems

Pythagorus, in the sixth century B. C., made acoustic experiments with monochords ("An acoustic instrument with a sounding box, one string and a moveable bridge." [15] ). While shortening the string of the monochord by 2/3, he raised the pitch by a fifth. By repeating this shortening he generated a series of pitches that, when reduced to one octave, formed the diatonic scale as shown in Example 1.

The diatonic scale of Pythagoras had perfectly tuned fifths, but, the major thirds were very sharp and the minor thirds were flat. For this reason thirds and sixths were thought of as dissonances up until the fifteenth century when the tuning system changed to just intonation. [16]

Example 1. Pythagorus's derivation of the diatonic scale.

Movement by perfect fifths

F - C - G - D - A - E - B

When reduced to one octave, this results in a diatonic scale

C - D - E - F - G - A - B - C

By using a second monochord simultaneously, Pythagorus also discovered that when he shortened the string by 1/2, he could ascend by octaves. When he ascended to equivalent pitches by octaves and fifths (seven octaves to twelve fifths), the resultant pitch with the octaves [17] was slightly lower than with the fifths. This discrepancy is an acoustic phenomena and tuning problem called the Pythagorean comma. This discrepancy was what later brought about the necessity of temperament as music became more harmonic. [18]

Aristoxemus in the fourth century BC, thought that the judgment of the ear more important than the math and protested against the rigidity of the mathematical theories. Aristoxemus had many scales, one of which was equal temperament. He suggested that since "pitch was a continuum it could be divided into equal intervals even if the mathematics of Pythagoreans could not express them as string lengths." [19] But, the scales he most used were ones that were equivalent to that of Pythagoras. [20]

Ptolemy in the second century AD, said that "tuning is best when the ear and ratios are in accord," [21] but the tunings must be superparticular ratios (where "the antecedent exceed[s] the consequent by 1, as 5/4, " [22] 4/3, 3/2, 2/1 etc.), a concept that today we regard as arbitrary. The intervals created by superparticular ratios include many, but not all, of the just intonation diatonic intervals, as shown in Example 2.

Example 2. Ratios of pure intervals, as derived from the numbering of partials in the overtone series. [23]

Ratio Interval

2/1 octave

3/2 perfect fifth

4/3 perfect fourth Superparticular ratios

5/4 major third

6/5 minor third

9/8 major second

16/15 minor second

5/3 major sixth

16/9 minor seventh

8/5 minor sixth

15/8 major seventh

Ptolemy presented, in his writings, twenty-one tuning tables [24] including the just intonation tunings (the Renaissance modification of Pythagorean tuning) of his own, and those of Archymas, Didymus and Eratosthenes [25] .

In the Middle Ages, the only ancient Greek tuning system known in detail, was that of Pythagorus. Boethius, in his writings, mentioned other Greek theorists and scales but presented the mathematical detail of only

Pythagorus. [26] Pythagorean tuning was best suited for melodies and was acceptable for Gregorian plainchant. Even with the addition of the fourths and fifths of organum, Pythagorean tuning was still adequate. As thirds and sixths became more common (in the fifteenth century), the common practice of tempering to soften the harshness of these intervals had already begun. [27]

Just intonation was adopted as the thirds and sixths became more widely accepted in the Renaissance. It was based on the harmonic series, combining the pure fifths (3:2) with the major thirds ratios (5:4), as shown in Example 3. The note E, for example could now be obtained directly from C

instead of ascending by fifths (C - G - D - A - E) through four operations

spanning over two octaves..

Example 3. Deriving the notes of just intonation from the interval of the third and the fifth.

E B F# C#

F - C - G - D - A - E - B

In Example 4 are the ratios of the scale produced in just intonation as compared to Pythagorean tuning. The ratios in just intonation are derived directly from the harmonic series.

Example 4. Comparing the intervals in just intonation with Pythagorean tuning.

C D E F G A B C

Just Intonation 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1 [28]

Pythagorean 1/1 9/8 81/64 4/3 3/2 27/16 243/128 2/1 [29]

In this way, some of the fifths were tempered slightly to accommodate the thirds. Just intonation can be thought of as an extension of Pythagorean tuning. [30] Generally, it is thought the simpler the ratio, the more consonant the interval. [31]

Nicola Vicentino was a sixteenth century theorist and composer. He advocated chromatic and microtonal music. He even built two instruments to play all of the ancient Greek genera. One of these was a "harpsichord with thirty-six keys to an octave called an archicembalo and the other a comparable

portative organ called an archiorgano" [32] Vicentino, in his writings, mentions the intonation problems that arose when instruments with different tunings played together (keyboard instruments used meantone temperament while fretted instruments like the viol and the lute used equal temperament). When unfretted violins became the backbone of the orchestra in the seventeenth century, this became less of a problem. [33]

Music theorists "Zarlino, Mersenne and Rameau presented just intonation as the theoretical basis for the scale but temperament was [thought to be] a practical necessity." [34] From the sixteenth through nineteenth centuries, many tempered tuning systems were developed. These tunings were called well temperaments and meantone temperaments and included tunings by Aron, Fogliano, Stevin, Werckmeister, Neidhardt, Young, Silbermann, Gallimard and others. Meantone temperament was regular (only one odd sized fifth) and restricted (modulations were limited to three sharps and two flats) [35] . The modulations were limited to three sharps and two flats because of the existence of so-called wolf tones. Wolf tones were notes that were so out of tune, they were thought to resemble the howling of wolves. [36] The enharmonic equivalents, G# and Ab for example, were so far distant that they could not be used in the same piece.

Well temperament was irregular (with more than one odd sized fifth [37] ) and unrestricted (had freedom of modulation). In well temperament, there is complete freedom of modulation but unlike equal temperament, the tones are not equally spaced and each key center has its own unique characteristics. [38] Well temperament and meantone temperament were practiced at the same period. Meantone temperament was favored when possible and until the eighteenth century well temperament was not as popular as meantone temperament. [39] . If more modulatory freedom was required, well

temperament was used. Mozart and early Beethoven used meantone temperament [40] while Bach, Chopin, and young Liszt used tunings in well temperament. [41]

The first definition of equal temperament was given by Salinas in 1577

regarding the placement of frets on the viol. [42] There were three mathematical methods inherited from the ancient Greeks to work out the placement of the frets [43] that we now understand to be based on logarithms. The first correct figures for equal temperament were published in China by the theorist Ling Lun in the twenty-seventh century BC [44] (although of no use for their music at the time). [45] Equal temperament is an arbitrary division of the octave into twelve equal parts. This distributes the Pythagorean comma equally amongst all the notes. The only pure interval ratios are those of the octaves. All the rest are compromises. Alexander Ellis in the nineteenth century developed a

system for comparing tuning methods with equal temperament as the

standard. Each octave was assigned 1200 cents. With twelve equal divisions to the octave, each semitone was equal to 100 cents. Now there was a method to compare the other systems. [46] Since the mid nineteenth century, equal temperament has been the dominant system in Western music. This solved the problem of modulation to keys with more than three sharps or two flats, and took care of enharmonic equivalents, but, it took away the distinctive characters of the different keys [47] and made all the music sound "equally gray." [48] Equal temperament came into ascendancy alongside developments in chromaticism, a lessening of interest in tonality, the iron frame piano and the rise of the professional piano tuner. Up to that point, musicians tuned their instruments by ear. Equal temperament was almost impossible to do this way as almost all intervals were slightly out of tune with the ear. [49] Additionally, the iron frame pianos, with the greatly increased string tension required a professional hammer technique to set the tuning pins. Due to the difficulty in re-tuning, an optimal tuning for all music had to be used. Musicians then no longer tuned their own instrument as they individually preferred, for

different pieces of music, but, deferred instead to the dictates of the professional tuner. [50]

The development of Western tuning systems can be summarized in Example 5.

Example 5. The Development of Western Tuning Systems

Pythagorean tuning

Just Intonation

Meantone Temperament Well Temperament

Equal Temperament

Equal Temperament Contemporary Temperaments

Equal temperament is an "acoustical compromise . . . designed to satisfy as completely as possible three incompatible requirements- true intonation, complete freedom of modulation and convenience in practical use in keyed instruments.- and that it sacrifices the first of these to the second and third." [51] In the ancient diatonic scale, the scalar steps are uneven, which promotes tonality. In the equal tempered scale, there are no beginnings and endings and no tonality. Modern music "promotes neutrality of key centers and abstract harmony. Equal temperament is the only completely atonal temperament that has existed in history." [52] " Resistance to equal temperament was not dead

until 1885. [53] Before the twentieth century, equal temperament, as we use it today, was not commonly practiced due to the technical limitations of the tuners, their instruments and ears. They used instead imitations now called quasi-equal temperament. [54] "Nineteenth-century temperament contained various degrees of 'key-coloration' which preserved the 'character of the keys.' Key-coloration has disappeared in today's tempering practice which is reinforced by the philosophy of even-tone voicing." [55]

Up until the universal adoption of equal temperament, tuning practices were quite varied. In his book, Tuning the Historical Temperaments by Ear, Owen Jorgenson lists tuning directions for twelve Pythagorean tunings, fourteen just intonation tunings, twenty-two meantone temperaments, forty-five well temperaments, fourteen rivals of equal temperament, and five contemporary tunings. [56] In his book Tuning and Temperament, Murray Barbour lists twenty-one Greek tuning tables, twenty-four just intonation tunings, seventeen meantone temperaments, and many others systems totaling one hundred and eighty different historical tuning

scales. [57] With our ears adjusted to equal temperament, we listen to music

from centuries past in temperaments other than in which the composer wrote. This produces "uniform modulatory indifference . . . with arbitrary shifts of registration at the harpsichord . . . instead of a tapestry of differentiated colors . . . and coloring of melody, harmony, modulation and voice leading as the composer wrote it." [58]

Twentieth Century Practices

In the twentieth century, despite the predominance of equal temperament tuning and the accompanying psycho-acoustic adjustment of our ears, many composers are using alternative tunings. Some performers have initiated temperament recitals where multiple pianos and harpsichords are onstage, variously tuned so that for each piece, the instruments can be played in the tunings that the composers wrote. [59] Composers using alternative tunings are often referred to as microtonalists and systematically use microtones in three ways; as embellishments in otherwise

equal temperament music, as smaller than semitone subdivisions of equal temperament (quarter-tones and eighth-tones) or as various, unequal subdivisions of the octave (extended just intonation). [60] It is with this latter

group that we will look more closely. Microtonalists that divide the octave unequally into twelve, nineteen, thirty-one, forty-three, fifty-three or more parts require new or revised instruments, new techniques of performance, modified ear training and adapted notational systems. [61] With the use of specifically tuned synthesizers [62] explorations have become more widespread. [63] To obtain the nineteen note scale, you use both versions of the black keys, for example A# and Bb are both used. [64] In a thirty-one note scale, you use all versions of all notes, C double sharp and D double flat also. For a greater number of notes to the octave, you add different adjustments for the different commas. [65]

Harry Partch lists his various musical influences as "Christian hymns, Chinese lullabyes, Yaqui Indian ritual, Congo puberty ritual, Cantonese music hall and Okies in a California vineyard." [66] Partch states that "originality cannot be a goal. It is simply inevitable." [67] Before he was twenty, he had rejected the intonation and concert systems of Europe [68] and set out to find his own intuitive path. [69] He began to "write music on the basis of harmonized spoken words, for new instruments and on new scales." [70] Partch called his music "corporeal, oral and visual." [71] To Partch, corporeal music involves "the whole body, the whole person, the whole mind." [72] "No work of music should be deprived of being also a work of theater, also a work of dance, also a work of literature, . . . a work of sculpture . . . and a work of architecture." [73] The ritualistic and dramatic are a large part of his work. Partch feels he has given as much time to imaginative and sculptural forms of instruments as to intonation. [74] His instruments are large, unusual and ornamented and are used onstage as a visual part of the performance. [75]

Partch created a concept that he termed monophony that was based on a few technical precepts. The ear perceives small number intervals (2/1, 3/2, 4/3) as consonances. [76] The higher the number ratio, the more dissonant the ear perceives the interval. Major and minor scales (Partch calls them Otonality and Utonality respectively) coexist in truly tuned diatonic scales. [77] Interval ratios of 3, 5, 7, 9, and 11 can be recognized as consonances. [78] (These were the intervals that he often worked with. The consonance of the eleventh is not universally recognized).

Partch was the center of his own musical universe. His music could only be played on his own instruments by performers specifically trained on them and can only be fully appreciated through a live performance. [79] Partch abandoned the quest for personal fame in the public or academia in favor of

"many isolated, localized, independent and unique ways of conceiving and practicing musical art." [80]

For Harry Partch, his scalar and intonational choices were the inevitable by products of his original and inventive way of approaching his art. He theorized about his choices and often used a specific forty-three note

scale. But, also, he would create instruments and find his scales from the sounds they would make. His flexibility was a central part of his creative approach and fundamental to his compositional process.

Ben Johnston was taught by Harry Partch, John Cage and Darius Milhaud. Partch always insisted that he wasn't a teacher and considered Johnston his apprentice. [81] Johnston beginning in the 1960's, began writing music in alternative tunings for conventional instruments, although much of it was using total serialism. [82] In his piece, Sonata for Microtonal Piano," Johnston states that "only seven of the eighty-seven keys on the piano have octave equivalents. Thus there are eighty-one different pitches, providing a piano with almost no consonant octaves." [83] During the 1970's, Johnston turned to "extended just intonation, which implies tonality." [84] Johnston uses an extended just intonation scale with fifty-three notes to an octave not only because he thinks it sounds better, "but that irrationally dissonant music, that is, the tempered music we hear every day, is physically and psychologically unhealthy for us." [85]

Johnston wanted "to compose music basically in the European tradition." [86] He believes that music causes psychological, political and social results and has a "causative effect on human relationships." [87] Music that "pretends to be in tune when it is not, is a bad moral model." [88] This what temperament is doing and that "the symbolism in music is appreciated by every human being on some level or another." [89] As an artist, Johnston doesn't want to deal with "those adulterated symbols because [he thinks] to some extent, they're poisonous." [90] Johnston states that "all of the dissonant music of the twentieth century may well be unhealthy for us. Not because its dissonant. Not even because its complex. But because its irrationally dissonant." [91] For Johnston, the cure to this problem requires people to recognize three things. First that music creates emotional states that don't just reflect what is already there. Secondly, people in our culture are involved with tonal music in one way or another. Thirdly, music is "not simply an intellectual exercise, it is a physical response to sound." [92]

Johnston feels that just intonation is a more complete answer and just about everything he has written since 1970 has been in extended just intonation. [93]

SUMMARY AND CONCLUSIONS

Music and its form have changed dramatically over the past few millennium; from homophonic melody, to triadic tonal harmony, to modulating and extended tonality, to the breakdown of tonality altogether. The accompanying systems of tuning have changed alongside of these developments in the music itself. Pythagorean tuning was good for predominantly melodic music while just intonation developed to support triadic harmony. Meantone and well temperament enabled composers to modulate and to extend the harmony. Equal temperament was best suited for the chromaticism of atonal, non-key-centered or serial music. [94] Today we have seen a return of interest in tonality and an accompanying resurgence of interest in older tunings by contemporary composers.

Contemporary composers using alternatives to equal temperament are called microtonalists. In this paper, we examined two of those using more than twelve unequal divisions of the octave called extended just intonation. This author concludes that both Harry Partch and Ben Johnston use alternatives to equal temperament tuning for ideological reasons that are central to their compositional process. They could not achieve their musical goals without using alternative tuning systems.

In listening to the music of the microtonalists, this author sees two distinct groups. Firstly, are those whose music sounds distinctly and consistently out of tune with our equally tuned ears. These composers use intervals in their compositions that are so far distant from our usual temperament that we, as listeners must adjust our musical judgment drastically to embrace the music as being acceptable. Both the music of Harry Partch and Ben Johnston fall into this category. The music has a distinctive character, and after considerable listening, begins to have a charm, and affability of its own. The second group of composers use alternative tunings to exploit the greater resonance of just intonation amongst instruments that are capable of dynamic tuning (the unfretted stringed instruments and some

of the wind and brass instruments). Of course, there is overlap between the two groups. At the MicroFest on April 19, 1998 at Pierce College in Woodland Hills, California, this author viewed the performance of music by eight contemporary microtonal composers (Lou Harrison, Harry Partch, Arvo Part, Johnny Reinhard, George Zelenz, Esmaeel Tehrani, and Sasha Bogdanowitsch). The work by Partch, December 1942, was the one that sounded most out of tune to this writer's ears. The piece by Tehrani, an Iranian composer, was performed by the composer on the santur in a Persian tuning. The tuning, although it overlaps with Western equal temperament at points, also has intervals that are extremely variant with it. The performance was very compelling as Tehrani was so focused in his performance that we in the audience, were enveloped by his reverie and unquestioningly accepted his notes as being the only ones that could be correct. In George Zelenz's Tiers of Yearning, you heard the resonance of pure intonation as the viola, cello, oboe, and contra-bass tuned to one another as they performed. In Arvo Part's Fratres, a cello drone played under a guitar with a custom fretboard that had infinitely adjustable frets for each string. This enabled the guitarist to set his scale for the piece to achieve a just intonation with the drone. Johnny Reinhard in a humorous and charismatic performance of his work entitled Dune, explored the limits and capabilities of the contra-bassoon. He demonstrated a wide range of timbres and tones that resembled at different times the clarinet, the oboe, the saxophone, and the donkey. He used techniques that expressed microtones, undertones, difference tones, multiphonics and the Doppler effect.

For future study, this author recommends listening extensively to works in alternative tunings and comparing the composer's intended scales with the psycho-aural results. Also she would like to undertake a comparative study involving contemporary composers and their notational solutions for alternative tuning. In summary, this author concludes that the use of alternative tunings by composers is an effective way to extend contemporary musical resources. And further, that the ideologies, practices and goals of the composers studied, necessitate their use of alternative tunings because their results could not have been achieved through the use of equal temperament.

SELECTED BIBLIOGRAPHY

Kennedy, Michael. The Oxford Concise Dictionary of Music, (Oxford: Oxford University Press, 1985), 651.

Barbour, Murray. Tuning and Temperament. (East Lansing: Michigan State College Press, 1951), 5.

Gaudentius, "Harmonic Introduction" in Source Readings in Music History (New York: W. W. Norton and Company, 1998), 74.

Jim Campbell, "The Equal Tempered Scale and Some Peculiarities of Piano Tuning" Precision Strobe Tuner. accessed April 10, 1998. Available from http://http://www.izzy.net/~jc/Temper.html; Internet.

Siegmund Levarie and Ernst Levy, Tone. (Kent, Ohio: The Kent State University Press, 1980), 224.

Owen Jorgenson, Tuning the Historical Temperaments by Ear (Marquette: The Northen Michigan University Press, 1977), 7.

Ian H. Henderson, Strobe Tuner Settings for the Historic Scales (New York: Henderson, 1983), 34.

Bill Alves, "Key Characteristics and Pitch Sets in Composing with Just Intonation." Journal of the Just Intonation1/1, 1990. accessed April 10, 1998. Available from http://www2.hmc.edu/~alves/justkeys.html; Internet.

William Morris, The American Heritage Dictionary of the English Language. (Boston: American Heritage Publishing Company, 1973), 848.

George Thaddeus Jones, Music Theory. (New York: Harper Perennial, 1974.

Bill Alves, " The Just Intonation of Nicola Vincentino." Journal of the Just Intonation Network 1/1, 5, No. 2, 1990. accessed April 14, 1998. Available from http://www2.hmc.edu/~alves/vincentino.html; Internet.

William Duckworth and Edward Brown, Theoretical Foundations of Music. (Belmont, California: Wadsworth Publishing Company, 1978), 19.

Chas Stoddard, A Short History of Tuning and Temperament.(Glastonbury: Stoddard, 1998) accessed April 26, 1998. Available from http://www- math.cudenver.edu/~jstarret/tuninghist.html; Internet.

Robert Rich and Carter Scholz, Just Intonation Calculator. (Soundscape Productions, 1997), Pythagorean tuning.

Llewellyn Lloyd and Hugh Boyle, Interval Scales and Temperaments (New York: St. Martin's Press, 1979),66.

Owen Jorgenson, Tuning (East Lansing, Michigan: Michigan State University Press, 1991), 2.

David Cope, New Directions in Music, sixth edition, (Madison: Brown and Benchmark, 1993), 67.

Paul Ehrlich, "Tuning, Tonality and Twenty-two-tone Temperament(Denver: Ehrlich, 1998),), accessed April 25\9, 1998. Available from http://www- math.cudenver.edu/~jstarret/Question1.html; Internet.

Harry Partch, Genesis of Music. second edition (New York: Da Capo Press, 1974), viii.

Harry Partch, "Barbs and Broadsides," edited by Danlee Mitchell and Jonathan Glasier. accessed April 26, 1998. Available from http://http://www.corporeal.com/bb_O.html; Internet.

John Struble, American Classical Music (New York: Facts on File, 1995), 278.

Derek Bermel, Ben Johnston. Paris New Music Review, ( Paris: Paris New Music Review Society, 1998), accessed April 25, 1998. Available from http://man104nfs.ucsd.edu/~hlivings/inter/johnston.html; Internet.

Phillip Bush, "Ben Johnston: Music For Piano" CD Liner notes ((New York: Koch, International Classics, 1996), 3.

Duckworth, Talking Music. (New York: Schirmer Books, 1995), 121.

Notes

[Note 1] Michael Kennedy, The Oxford Concise Dictionary of Music, (Oxford: Oxford University Press, 1985), 651.

[Note 2] Barbour, Tuning and Temperament. (East Lansing: Michigan State College Press, 1951), 5.

[Note 3] Gaudentius, "Harmonic Introduction" in Source Readings in Music History, edited by Thomas J. Mathiesen(New York: W. W. Norton and Company, 1998), 74.

[Note 4] Gaudentius, 74.

[Note 5]

Jim Campbell, "The Equal Tempered Scale and Some Peculiarities of Piano Tuning" Precision Strobe Tuner. accessed April 10, 1998. Available from http://http://www.izzy.net/~jc/Temper.html; Internet.

[Note 6] Campbell, 2.

[Note 7] Barbour, Tuning and Temperament. 5.

[Note 8] Siegmund Levarie and Ernst Levy, Tone. (Kent, Ohio: The Kent State University Press, 1980), 224.

[Note 9] Owen Jorgenson, Tuning the Historical Temperaments by Ear (Marquette: The Northern Michigan University Press, 1977), 7.

[Note 10] Ian H. Henderson, Strobe Tuner Settings for the Historic Scales (New York: Henderson, 1983), 34.

[Note 11] Henderson, 34.

[Note 12] Henderson, 34.

[Note 13] Henderson, 34.

[Note 14] Bill Alves, "Key Characteristics and Pitch Sets in Composing with Just Intonation." Journal of the Just Intonation1/1, 1990. accessed April 10, 1998. Available from http://www2.hmc.edu/~alves/justkeys.html; Internet.

[Note 15] William Morris, The American Heritage Dictionary of the English Language. (Boston: American Heritage Publishing Company, 1973), 848.

[Note 16] Barbour, 1.

[Note 17] "An interesting consideration is the phenomenon of the octave. Why is it, when we consider the audible frequency range from 20 Hz to 20 KHz, we perceive a series of points along this scale that we can consider as having the same "quality" while patently being a different note? Part of the explanation may be that if we take a bi-lateral cross- section through the cochlea, that part of the ear's mechanism responsible for converting acoustic energy into electrical impulses, it reveals a spiral shape which can be described mathematically by a Fibonacci Series; the same maths govern the principles of the harmonic series. Neuro-pathology of the ear shows that octaves are decoded at the same point in each layer of the spiral. Some experts maintain that if the cochlea was a straight cone, rather than a tightly-wound spiral, we would have no perception of the octave at all; all we would hear would be a series of successively rising tones." Chas Stoddard, "A Short History of Tuning and Temperament" (Glastonbury: Stoddard, 1998) accessed April 26, 1998. Available from http://www-math.cudenver.edu/~jstarret/tuninghist.html; Internet.

[Note 18] George Thaddeus Jones, Music Theory (New York: Harper Perennial, 1974.), 40.

[Note 19] Bill Alves, " The Just Intonation of Nicola Vincentino." Journal of the Just Intonation Network 1/1, 5, No. 2, 1990. accessed April 14, 1998. Available from http://www2.hmc.edu/~alves/vincentino.html; Internet.

[Note 20] Barbour, 2

[Note 21] Barbour, 2.

[Note 22] Barbour, xii.

[Note 23] Henderson, 18.

[Note 24] Barbour, 2.

[Note 25] Alves, "Vincentino," 2.

[Note 26] Barbour, 2.

[Note 27]

Barbour, 3.

[Note 28] Levarie, 415.

[Note 29] Robert Rich and Carter Scholz, Just Intonation Calculator. (Soundscape Productions, 1997), Pythagorean tuning.

[Note 30] William Duckworth and Edward Brown, Theoretical Foundations of Music. (Belmont, California: Wadsworth Publishing Company, 1978), 19.

[Note 31] Stoddard, "A Short History of Tuning and Temperament."

[Note 32] Alves, "Vicentino," 1.

[Note 33] Barbour, 8.

[Note 34] Barbour, 11.

[Note 35] Barbour, x.

[Note 36] Barbour, 11.

[Note 37] Barbour, x.

[Note 38] Barbour, 2.

[Note 39] Jorgenson, Historical Temperaments, 7.

[Note 40] Jorgenson, Historical Temperaments, xiii.

[Note 41] Jorgenson, Historical Temperaments, xii, 3.

[Note 42] Barbour, 6.

[Note 43] Barbour, 53.

[Note 44] Jorgenson, Historical Temperaments, 3.

[Note 45] Barbour., 7.

[Note 46] Levarie, 224.

[Note 47] Duckworth, Foundations , 20.

[Note 48] Duckworth, Foundations, 294.

[Note 49] Jorgenson, Historical Temperaments, 12.

[Note 50] Jorgenson, Historical Temperaments, 12.

[Note 51] Llewellyn Lloyd and Hugh Boyle, Interval Scales and Temperaments (New York: St. Martin's Press, 1979),66.

[Note 52] Owen Jorgenson, Tuning (East Lansing, Michigan: Michigan State University Press, 1991), 2.

[Note 53] Jorgenson, Tuning, 3.

[Note 54] Jorgenson, Tuning, 3.

[Note 55] Jorgenson, Tuning, 3.

[Note 56] Jorgenson, Historical Temperaments, xxv.

[Note 57] Barbour, 184.

[Note 58] Barbour, xii.

[Note 59] Barbour, 2.

[Note 60] Duckworth, Foundations. 293.

[Note 61] David Cope, New Directions in Music, sixth edition, (Madison: Brown and Benchmark, 1993), 67.

[Note 62] Cope, 66.

[Note 63] Paul Ehrlich, "Tuning, Tonality and Twenty-two-tone Temperament(Denver: Ehrlich, 1998),), accessed April 25\9, 1998. Available from http://www- math.cudenver.edu/~jstarret/Question1.html; Internet.

[Note 64] Duckworth, Talking Music, 130.

[Note 65] Duckworth, Talking Music, 130.

[Note 66] Harry Partch, Genesis of Music. second edition (New York: Da Capo Press, 1974), viii.

[Note 67] Partch, Genesis, xi.

[Note 68] Partch, Genesis, vi.

[Note 69] Partch, Genesis, 6.

[Note 70] Partch, Genesis, 6.

[Note 71] Harry Partch, "Barbs and Broadsides," edited by Danlee Mitchell and Jonathan Glasier. accessed April 26, 1998. Available from http://http://www.corporeal.com/bb_O.html; Internet.

[Note 72]

Partch, "Barbs," 2.

[Note 73] Duckworth, Talking Music, 137.

[Note 74] Partch, "Barbs," 4.

[Note 75] John Struble, American Classical Music (New York: Facts on File, 1995), 278.

[Note 76] Partch, Genesis, 87..

[Note 77] Partch, Genesis, 89.

[Note 78] Partch, Genesis, 92.

[Note 79] Struble, 279.

[Note 80] Struble, 280.

[Note 81] Derek Bermel, Ben Johnston. Paris New Music Review, ( Paris: Paris New Music Review Society, 1998), accessed April 25, 1998. Available from http://man104nfs.ucsd.edu/~hlivings/inter/johnston.html; Internet.

[Note 82] Phillip Bush, "Ben Johnston: Music For Piano" CD Liner notes ((New York: Koch, International Classics, 1996), 3.

[Note 83] Bush, 7.

[Note 84] Bush, 4.

[Note 85] Duckworth, Talking Music. (New York: Schirmer Books, 1995), 121.

[Note 86] Duckworth, Talking Music, 138.

[Note 87] Duckworth, Talking Music, 148.

[Note 88] Duckworth, Talking Music, 150.

[Note 89] Duckworth, Talking Music, 150.

[Note 90] Duckworth, Talking Music, 150.

[Note 91] Duckworth, Talking Music, 151.

[Note 92] Duckworth, Talking Music, 154.

[Note 93] Duckworth, Talking Music, 156.

[Note 94] Duckworth, Talking Music, 150.