THE EVOLUTION OF HARRY PARTCH'S TUNING SYSTEM
by
B. McLaren
[with some corrections by Joseph Monzo]
Important questions in Partch scholarship remain unanswered:
[1] What catalyzed Partch's decision to abandon 12 tone equal temperament?
[2] Did Partch build instruments prior to 1925-1926? If so, what were they?
[3] Do all the pages except for the title and final page of the 1933 draft of Exposition Of Monophony date from 1933? If some of the pages date from earlier drafts, what are they--and from which draft do they date?
[4] What were the ratios of Partch's initial just array of 1923? What thought process led him to expand his initial 12-tone just array up to 55 tones, and then back down to 41 between 1923 and 1933?
[5] When and how did Partch arrive at the concept of the Tonality Diamond?
[6] Why did the Exposition of Monophony go through 5 drafts between 1927 and 1933?
[7] Why was there a three-year gap between the first and second drafts of the Exposition (1927 to 1930) while the subsequent three drafts took place at one-year intervals?
[8] What was the organization and what were the general contents of the original 1928 Exposition?
[9] Why did the 1927-28 Exposition have 90 pages, while the 1933 Exposition has only 54?
This paper proposes answers to these questions based on documents written by Partch himself. In some cases the conclusions are based on forensic methods (i.e., examination of pages of the 1933 Exposition under black light to retrieve original ratios whited-out and inked over with new ones), and in some cases on guesstimates and circumstantial evidence. Thus, some of the answers proposed here stand on firm ground, while others remain highly speculative.
1. THE ORIGIN OF PARTCH'S INTEREST IN JUST INTONATION
On pages 50 and 51 of the 1933 Exposition of Monophony Partch states:
In the spring of 1923 I ran across a book in the Sacramento Public Library, the name of which I have forgotten, on the relation of the physics of sound to musical theory, and containing a discussion of the merits of just intonation and equal temperament. That was the beginning of my present work. [Partch, 1933 draft of Exposition of Monophony, page 50]
Was this book Helmholtz's On The Sensation of Tone? Partch gives us a hint: on page 51, he writes:
My early work was not influenced by personal contacts; the later period was, casually. Several books have helped mold the work, particularly Helmholtz' Sensations of Tone. [sic]] [Partch, 1933 draft of Exposition of Monophony, page 51]
Yet Partch was meticulous. Would he state on page 50 that he ran across a book whose name he had forgotten, then on page 51 specifically name the book?
The evidence from Partch's own writings is unclear. On page vii of Genesis Of A Music (2nd ed., 1974), he states
When I was twenty-one I finally found, in a library, the key for which I had been searching, the Helmholtz-Ellis work, On the Sensations of Tones. Under this new impetus, doubts and ideas achieved some small resolution, and I began to take wing.
Partch would still have been twenty in April, 1923, since he was born in June [Note from Monzo: McLaren is in error here. Partch's 20th birthday was June 24, 1921. In April 1923 he was 21.]. Thus Harry's memory is playing a small trick on him here. But he would have been very nearly twenty-one in April 1923 [from Monzo: As pointed out above, Partch was already 21 in April 1923, 2 months before his 22nd birthday], so it's entirely possible that Helmholtz was the source of his inspiration. Moreover, he states on page vi of the Preface To the Second Edition of Genesis Of A Music:
Before I was twenty, I had tentatively rejected both the intonational system of modern Europe, and its concert system... In 1919, as I recall, I had virtually given up on both musical schools and private teachers, and had begun to ransack public libraries...
In 1919 Partch would have been 17 or 18, depending on the exact date. Dayton C. Miller's The Science of Musical Sounds had been published 2 years earlier, in 1916 [from Monzo: Dayton's book was published 3 years earler]. In the process of "ransacking public libraries," Partch would almost surely have encountered it.
This suggests a chain of events. The initial influence which broke Partch free from the 12-tone scale might have been Dayton Miller's book--not the Helmholtz/Ellis text. That tome might have provided the definitive kick that sent him in the direction of just intonation.
Examination of the bibliography to the 1974 Genesis of A Music turns up only one book other than Sensations of Tone dealing with physics and music and published prior to 1923: Miller, Dayton C. The Science of Musical Sounds. MacMillan, New York. 1922. [sic: the actual date of publication was 1916] [from Monzo: The 1st edition of Miller's book was published in 1916; the 2nd edition was published 1922.]
The Miller text is a remarkable work. Using an ingenious mechanical Fourier analyzer, Miller was able to obtain continuous graphic records of the waveforms of musical instruments as early as 1905. From these he calculated the waveforms' spectra by Fourier analysis. The book includes a fascinating discussion of ways to identify and correct errors in the transcript from the phonodeik, an early-1900s mechanical version of the much later oscillograph.
Partch writes in Genesis of A Music:
Sometime between 1923 and 1928 I finally became so dissatisfied with the body of knowledge and usages as ordinarily imported in the teaching of music that I refused to accept, or develop my own work on the basis of, any part of it. [Partch, Genesis of A Music, 2nd edition, 1974, page 4]
Why did Partch become "dissatisfied" with music as it was taught in the schools in the teens and twenties of this century?
On page 141 of Genesis of A Music, he reproduces "HARMONIC CURVES RECORDED ON MILLER'S PHONODEIK." (These tracings are what we today would call oscillograms: they depict the amplitude of sound waves as a function of time.) Partch cites Miller's 1937 Sound Waves: Their Shape and Speed as the source of these illustrations, but in fact identical illustrations can be found in Miller's earlier 1916 tome.
The tracings from the phonodeik must have had a powerful effect on Partch. Here, for the first time, he could see sound waves and count the number of wavecycles that defined a musical interval.
Could this be the cause of Partch's dissatisfaction? Having seen the sound waves for himself in Miller's phonodeik illustrations, it must have been difficult for him to believe that irrational ratios like 27/12 constituted a reasonable alternative to the much simpler and physically straightforward 3:2. As Partch sat in the Sacramento Public Library, leafing through either the Helmholtz or the Miller text, he must have grown more and more appalled at the disjunction between music--as it was then taught in public schools--and the physical basis of the sounds he was hearing.
Even today, Miller's book makes a powerful effect. But in 1923 the impact must have been revelatory.
From the back of the envelope in which the 1933 Exposition was mailed, we know the date of Partch's epiphany: April, 1923.
This envelope contains a very nearly complete record of my work in the musical system of Monophony covering ten years, April, 1923, to August, 1933...
It seems likely, therefore, that Partch's initial impulse toward just intonation derived from his encounter with either Helmholtz's On The Sensations Of Tone or Dayton Miller's The Science of Musical Sound in the Sacramento public library in April, 1923. At this remove, however, it is impossible to say which text he read first, or which affected him more. Both show a powerful influence in the later Genesis Of A Music: both authors are quoted.
2. DID PARTCH BUILD INSTRUMENTS PRIOR TO 1925-1926?
Was the Adapted Viola Partch's first xenharmonic instrument?
In the program notes to Water! Water!, Partch writes:
I began theorizing about new instruments in 1923, making small experiments in 1924, and building them in 1928. [Partch, notes to Water! Water!, 1961, last page]
Partch must have begun thinking about new instruments after his discovery of Miller's and Helmholtz's texts. Clearly the piano and other conventional instruments were inadequate to the task of producing just intonation... Therefore new instruments were needed. But what kind of instruments?
Again, Dayton Miller's The Science of Sound suggested a direction. Partch quotes this passage from Miller's book in Genesis of A Music, on page 96:
The science of sound should be of inestimable benefit in the design and construction of musical instruments, and yet with the exception of the important but small work of Boehm in connection with the flute, science has not been extensively employed in the design of any instrument. This can hardly be due to the impossibility of such application, but rather to the fact that musical instruments have been mechanically developed from the vague ideas of the artist as to the conditions to be fulfilled. When the artist, the artisan, and the scientist shall all work together in unity of purpose and resources, then unsuspected developments and perfections will be realised... [Miller, Dayton C., The Science of Musical Sounds, pp. 263-265, MacMillan, New York. 1916]
Eight pages later, In Genesis, Partch lets slip further evidence of his earliest instrument-building experiments:
The cello has several features to recommend it as an experimental instrument aside from its well-known musicality. With the bow, sustained tones in double stops are possible; the eyes are normally close to the fingerboard in playing; there is plenty of room for small-interval marks on the long fingerboard; and pinheads or bards or other indicators are easily hammered into holes in the fingerboard which has been prepared with an awl. Instead of inserting pinheads or brads immediately, however, it is perhaps a good plan to wrap a thickness of heavy paper around the fingerboard and paste it securely; marks on this will be susbtituted for pinheads, and in this way the investigator can experiment until he is sure what he wants. Also, a bridge lower than that ordinarily used, to decrease the extra tension on the strings when high stops are made, is advisable. [Partch, H., Genesis of A Music, 1974, 2nd edition, page 99]
Most significant of all, on page 41 Partch has originally typed 'CELLO ADAPTATION. This is scratched out in No. 2 pencil and written over with the word STRING FAMILY instead of 'CELLO.
The picture seems clear. After arriving at the idea to use just intonation, in 1924 Partch acquired a cast-off cello and taped a piece of paper to the fingerboard. He began to mark off string lengths in small whole number ratios. Elsewhere in Genesis, Partch suggests tuning up several of the strings of a Harmonic Canon to compare equal temperament and just intonation:
A harmonic canon is a simple and direct way of beginning an investigation into tonal resources...
Almost certainly this was one of the first experiments he tried on the cello fingerboard. Were the differences between just intonation and equal temperament audible? Yes--they were. Could just intervals be played reliably on the cello? Clearly, they could.
However, the 1924 cello adaptation posed problems for an itinerant composer:
The main disadvantages of the cello are in part the same as those of the harmonic canon. Non-uniformity in strings and in bowing technique, increase of tension in stopping, inexact stopping, and the almost unconscious use of vibrato, etc., are factors not conductive to a really adequate intonational investigation. But with all its faults, the experimental cello nevertheless represents an immediately feasible beginning, the end result of which might be unpredictably fruitful. [Partch, H., Genesis of A Music, 2nd ed., 1974, pg. 99]
Just as important, a cello is bulky. In 1923-1924 Partch had to get around by trolley and on foot. A cello is a hefty piece of cartage to lug around: you can't run for a bus with it. Something else was needed--something more portable.
The violin and viola suggest themselves immediately. Like the cello, these instruments are unfretted and lend themselves naturally to a variety of intonation. And Partch again gives us evidence that he experimented with a violin and a viola fingerboard in just intonation between 1925 and 1926. In Genesis of a Music, page 488 (2nd edition, 1974), he writes:
1925-1926 Experiments begun in San Francisco with paper coverings for fingerboards of a violin and a viola. Marking for Just Intonation, mathematically determined, were on the coverings.
But the problem with a violin or a viola fingerboard is that it's not easily visible to the performer when played on the shoulder in the traditional way--and the fingerboard is much shorter than that of the cello. What to do?
Partch reached a compromise in 1928 when he removed the fingerboard from the cello and had it grafted on the body of a discarded viola in 1930. He played the instrument like a cello, because "If the instrument is a violin or viola it should be held between the knees when played, to more accurately make the stops." [Partch, H. Exposition Of Monophony, 1933, pg. 6]
Finally...something portable, inexpensive and practical. Something that worked.
Tantalizingly, Bertha McCord Knisely writes in December 1933 that Partch "is in New York pursuing his objective--the completion of the keyboard" of which Partch built a model in April, 1932. She goes on to note that "Meanwhile he has worked on the adaptation of a cello and is already engaged in writing monophonic music for three stringed instruments." [Knisely, B., Los Angeles Saturday Night, December 30, 1933, pg. 12]
This is hearsay evidence and thus suspect. Did she get it right? Had Partch adapted three string instruments to Monophony, rather than just one, by December 1933? Or was she confusing his previously adapted cello with the viola, whose fingerboard came from the earlier cello adaptation?
It's impossible to say. No trace remains of an adapted cello produced later than 1933, and no trace remains of Partch's compositions for three adapted string instruments--if any such music existed. We know Partch composed a string quartet in just intonation in 1925: the preface to the second edition of Genesis Of A Music tells us so. "About 1925 I wrote a string quartet in Just Intonation and compiled a set of Just-Intonation resources soon afterward." [Partch, Harry, Genesis of A Music, vii, 2nd ed., 1974]
The close proximity in time between Partch's just string quartet and his compilation of the first 1927-28 draft of the Exposition of Monophony strongly suggests that Partch developed the 29 pitches of his first Monophonic fabric for the quartet and codified them in his Exposition. Thus there is clear precedent for Partch to compose in just intonation for string instruments.
Moreover, Mrs. Knisely mentions that Partch is writing the music for adapted string instruments (3, not 4) in December, 1933--not earlier, in 1925. Therefore the putative Partch string trio of 1933 is a different work from the lost 1925 Partch just intonation string quartet. Lastly, page 41 of the 1933 Exposition contains the heading 'CELLO ADAPTATION scratched out and replaced by the handwritten note STRING FAMILY ADAPTATION. This is an unmistakable indication that in 1932 when page 41 (originally numbered 72) was originally written, and as late as 1933, Partch intended to produce an entire family of unfretted string instruments adapted to Monophony.
Did Partch complete or even begin a string trio for Monophony? Or was it an idea he pursued without results?
This remains a knot for future scholars to untangle.
3. DO PAGES OF THE 1933 EXPOSITION DATE FROM EARLIER DRAFTS?
Both the title and final pages (the latter numbered 90 in india ink and written by hand in India ink rather than typed) are known to date from 1927-28. Partch tells us so in his own handwriting. Above the title , Partch has written in pencil: "These are the first and last pages of "Exposition of Monophony" -- notarized." The title page is dated in India ink May 20, 1928; the date of notarization is May 26, 1928.
Thus there is no question that the first and last pages date from 1927-28.
The 1933 draft contains 54 pages: the unnumbered title page, the unnumbered page 2, marked PREFACE, numbered pages 1-51, and page 54, numbered 90 in India ink. The 1933 draft contains two separate histories of the manuscript: on unnumbered page 2, marked PREFACE, Partch has written in pencil at the top:
This envelope contains a copy of the 1933 draft of "Exposition Monophony," (also 2 pages of 1928 draft draft notarized) and a more or less complete record of the evolution and presentation of the system "Monophony" and the music written in it -- Also, three dated designs of Ratio Keyboard, the 2nd notarized -- Also pictures of charts, adapted viola and ratio keyboard model (Pages 37, 49).
In the left margin, Partch has pencilled:
"Exposition of Monophony" was written and rewritten, completed at the following times and places:
May 20, 1928
1666 Clay St.
San Francisco
Oct. 21, 1930
828 Camp St.
New Orleans
Oct. 2, 1931
175 6th Street
San Francisco
Aug. 24, 1932
Visalia, Cal.
June 8, 1933
327 So. Hope St.
Los Angeles
However, while Partch makes it clear that the first and last pages date from 1928, he never specifies that all the pages from the fifth draft date from 1933. As will be seen, there is strong reason to believe that a number of pages from the 1933 draft were recycled from earlier drafts.
The first piece of evidence comes from the numbering of the pages.
All of the pages of the 1933 Exposition are numbered in pencil at the upper right hand corner, except for the unnumbered title page and the second page, marked PREFACE, and the last page.
Four of the pages, however, originally had other numbers--in India ink. These numbers have been scratched out with pencil but are still visible.
Pages 17 and 18 were originally marked 18 and 19 (respectively) in India ink. The original India ink page numbers, identical to the calligraphy on page 90 from 1928 and written in the same kind of ink with the same kind of pen, have been scrawled out and written over in pencil--the same thick No. 2 pencil used to number the rest of the pages. Moreover, the texture and bond of the paper used on pages 18 and 19 is entirely different from that of subsequent or previous pages: and Partch uses red ink to indicate successive subdivisions of string lengths--red ink is used nowhere else in the Exposition, except for the ruling lines drawn on page 22 and on page 36.
There is additional evidence that not all pages of the 1933 Exposition date from that year. Pages 1, 22, 25 and 41 were all written on a Pica typewriter. using faded carbon paper. However, all the other pages of the 1933 Exposition are written on an Elite typewriter with a new, crisper piece of carbon paper.
Pages 22, 25 and 41 appear to have been written with the same typewriter and carbon paper as the front page, known to date from 1928: a typewriter very different from that used on all the other pages.
More: pages 17, 18, 22, 25 and 41 were originally numbered in India ink (respectively) 18, 19, 54, 55, and 72--page 36 appears to have always been numbered 36 (also in India ink); there is no new page number penciled over it.
This provides a third piece of evidence: because the scratched-out India ink numbers go up to 72, while the last page of the 1928 Exposition is numbered 90--also in India ink. Yet there are only 51 numbered pages in the 1933 edition of Exposition Of Monophony, all enumerated in pencil.
The only possible conclusion is that Harry recycled some of the pages from an earlier edition of the Exposition.
Why would he do so?
An examination of the extremely elaborate pages 17, 18, 25 and particularly the large page 36 with a complete interval array on it provides a possible answer: there's a huge amount of work invested in these pages. Many hours of painstaking drawing with India ink--using different colors of ink, both black and red.
Is it possible that these pages date from the original 1927-28 edition of the Exposition?
At this point, the trail of physical evidence stops. Thus we must resort to circumstantial evidence.
Partch's ratio keyboard diagram is known to date from April 1932--the notarization seal and date are affixed to the diagram, and the enclosed letter gives the set of ratios of Partch's scale at that time.
These 39 ratios are identical to the pitches found on page 41 of the 1933 Exposition.
Reviewing the evidence, we note that page 41 was originally numbered 72 in India ink; it uses a different typewriter from the one used to type the 1933 pages of the Exposition: and it bears the heading CELLO ADAPTATION, subsequently scrawled out in pencil and revised to read STRING FAMILY ADAPTATION. [from Monzo: Partch did not type these words in all capitals, capitalizing only the first letters and underscoring both words with the typewriter; and he did not cross out "Adaptation" but only "'Cello", hand-writing "String Family" to the left of "'Cello".]
Occam's razor tells us that page 41 (nee 72) was originally written around April, 1932. According to Partch's pencilled chronology of the Exposition, this means that page 41 of the 1933 Exposition was actually recycled from the third draft.
There is also good reason to believe that page 22 (originally numbered 54 in India ink) dates from the third draft. On page 51, Partch states:
"The 1931 draft admitted undertones as the source of intervals. This was the result of hastened thru reading Henry Cowell's "New Musical Resources."" [The words "the result of" were originally typewritten and have been scrawled out with a pencil; the words "hastened thru" are written over the old phrase.]
This is a crucial piece of evidence. It tells us that Partch did not use the tonality diamond until the 1931 draft--since the Tonality Diamond generates intervals both from undertone and from overtone series. By Partch's own testimony, therefore, page 22--containing a diagram of the Tonality Diamond--probably also dates from the third draft of the Exposition.
According to the same line of reasoning, page 18 (originally numbered 19 in India ink) must also date from the third draft in 1931, since it diagrams the undertone series. The handwriting and layout of page 17 (originally numbered 18 in India ink) are so similar that it seems likely Partch produced the page at the same time. Thus Page 17 probably also dates from the third 1931 draft.
This leaves only pages 25 and 36 unaccounted for.
These pages are both unusual in that they consist of new writing pasted over older writing. Using a black light placed behind the manuscript, the original writing can in both cases be discerned.
On Page 25, the old Pica typewriter (different from the one used in the 1933 Exposition) has written underneath a pasted-over portion of manuscript:
"[To prove this phenomenal causation] The other series may be taken from this series in their [sic] pristine form For example, the 5th overtone (5, 10, 20, 40, 80. is the series of likenesses that becomes the anchor of relation that creates the new series) (Similar intervals of graph are of the same width. That is, the 2/1 for example, 1 to 2, 2 to 4, 4, to 8, 5 to 10, 10 to 20, etc, are given the same paper extent)[ Partch, H., Exposition of Monophony, page 25, revealed by black light examination: words "To prove this phenomenal causation" written in India ink above the typewritten sentence]
This is a subdivision of the overtone series, not the undertone series. Thus page 25 may date from the 1930 or even the 1928 draft of Exposition.
Page 36 has also been altered. With a series of pasted-over folds, it shows signs of having been augmented. The page now measured 18.75 inches by 11. No such size of paper is manufactured int he United States, to my knowledge; the simplest explanation is that page 36 was originally a 17 x 11 inch page containing a version of Partch's scale dating from before the June 1933 fifth draft of the Exposition. The added length was produced by subsequent pasted-on overlays to expand the scale.
8 new pitches have been added-- two at either end of the scale, where secondary tones are most needed to avoid large gaps. If Partch's original 29 tone scale from 1928 had 8 secondary tones added to it, this would give a total of 37 tones--the same number of pitches specified in the 1933 Exposition.
Thus it seems likely that page 36 (by far the most elaborate of all the pages) dates from the original 1927-1928 or the second 1930 edition.
Another piece of evidence can be found on page 36. The classification of intervals is different from that used in the 1933 draft. On page 34, a pencilled diagram shows the intervals along with Partch's description: the 3-limit intervals are identified with POWER, the intervals at the extreme ends of the scale are identified with APPROACH, the intervals clustered around the 5/4 and 6/5 are identified with EMOTION and the intervals between the 4/3 and 3/2 are classified as SUSPENSE INTERVALS.
However, on page 36 Partch has originally written PSYCHIC INTERVALS instead of SUSPENSE INTERVALS. This description is scratched out--using India ink--and the familiar description INTERVALS OF SUSPENSE is written in below the original PSYCHIC INTERVALS.
On page 50 Partch states that the first draft of the Exposition in 1928 "contained a similar classification of intervals." If page 36 does indeed date from 1927-1928, this explains what he means: he originally used the term "PSYCHIC," instead of "SUSPENSE" to describe the intervals between 4/3 and 3/2.
The proposed dating of pages is therefore:
Unnumbered title page - 1927-8, 1st draft
Unnumbered preface through page 16 - 1933, 5th draft
Pages 17-18 - 1931, 3rd draft
Pages 20-21 - 1933, 5th draft
Page 22 - 1931, 3rd draft
Pages 23-24 - 1933, 5th draft
Page 25 - Unknown: could be 1927-8, 1930, 1931 or 1932
Pages 26-35 - 1933, 5th draft
Page 36 - 1927-8, 1st draft
Pages 37-40 - 1933, 5th draft
Page 41 - 1931, 3rd draft
Pages 42-51 - 1933, 5th draft
Last page (numbered 90 in India ink) - 1927-8, first draft
This chronology of pages is based on circumstantial evidence and is thus speculative. While it is clear that at least 6 of the pages of the 1933 Exposition date from earlier draft , it is impossible to be absolutely sure of their exact provenance. The probabilities favor this chronology--but it could be wrong.
4.WHAT WERE PARTCH'S ORIGINAL JUST PITCHES IN 1923?
To this point we've had hard evidence to go on. But now, time sweeps dust across Harry's path. The trail of evidence is harder to follow. No record exists in Partch's hand of an original just intonation scale from 1923. The closest we have is the scale which, on page 37 of the 5th 1933 draft of Exposition of Monophony, Partch describes thus:
The fingerboard of the viola was begun in Santa Rosa, California in 1928 and attached by Edw. Bentin in New Orleans in April, 1930. There are 29 indicators for ratios within the 2/1 (octave) corresponding to my 1928 theory of the more essential tones. [Partch., H., Exposition of Monophony, page 37, 1933 draft]
When he says "the fingerboard of the viola was begun," Partch almost certainly means that he began to use an awl to bore holes and a hammer to pound brads into the wood. He strongly suggests elsewhere in Genesis (page 99) that he had experimented with a cello fingerboard (used for the Adapted Viola) much earlier.
Clearly Partch had arrived at a just scale by 1923. What was it?
Partch discovered the Helmholtz/Ellis Sensations of Tone before or at the age of 21. He made the leap to just intonation. He marked off the cello fingerboard and tried out ratios.
Which ratios? Into how many tones does he first divide the 2/1?
No definitive record exists, but there is fragmentary evidence for the size of Partch's scale. He states in Exposition Of Monophony that "The following five years [after 1923] were a period of gestation. At different times I invented types of notation for 12 just tones to the 2/1, and wrote music for such to be played on unfretted string instruments." [Partch, H. Exposition Of Monophony, Page 50, 1933 draft]
Bertha McCord Knisely, to whom the August, 1933 draft of the Exposition was given for "thoughtful criticism," (according to page 51 of the Exposition) writes on page 12 of the December 30, 1933 edition of Los Angeles Saturday Night:
We wrote about Harry Partch's monophony, in these columns, many months ago. We told of his scale of thirty-seven intervals (he had worked it out to fifty-five but decided, later, on a less formidable development for immediate use) and of his adaptation of a viola for practical application of his theory. Also we told of a keyboard he had constructed representing three hundred-odd intervals for which he had secured a tone-producing mechanism." [Knisely, B., Los Angeles Saturday Night, pg. 12, Dec. 30, 1933]
(The reference to "three hundred-odd intervals" is confusing until it becomes clear that Ms. Knisely must be referring to the 8 octaves of 37 just pitches per octave. Clearly she means the entire gamut of 8*37 [= 296] pitches, rather that some previously-unknown 300-note just division of the octave.)
This gives us hard numbers: Partch used as few as 12 and as many as 55 just pitches to the 2/1 between April, 1923 and August, 1933 before settling on 37 notes.
It seems reasonable to deduce that Partch started with 12 just pitches, expanded the scale to 55 pitches, then shaved it back down to the 37 presented in the 1933 Exposition.
If so, with which 12 just pitches did Partch begin?
Given the powerful influence of the Helmholtz/Ellis work, is it possible that Partch began with the central pitches of Ellis' duodenarium?
It's impossible to say. Any such speculation is likely futile--no record exists of Partch's 12-tone just scale.
However, there is a record of an 11-tone just scale.
On page 8 of the 1933 Exposition, Partch gives an 11-tone scale derived by inverting string lengths (the frequency of a note sounded by a vibrating string is inversely proportional to its length): For strings lengths of 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9 and 1/15 with a notation "(15 inches long)," Partch derives the following scale:
Major
C D E F G A B
1/2 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 15/8 16/9
Notice that Partch starts on C in this scale. Is this a vestigial remnant of his earliest scale, from a period before he settled on 392 Hz G as his 1/1?
Again, it's impossible to say.
However, note that Partch's 11-tone scale differs in only one pitch from the central 12 notes of Ellis' duodenarium--Partch leaves out Ellis' 45/32 tritone:
ELLIS:
1/1 16/15 9/8 6/5 5/4 4/3 3/2 8/5 5/3 15/8 9/5
Thus it's tempting to speculate that Partch began with a set of essentially diatonic pitches derived from Ellis' duodenarium.
However, Partch would quickly have realized that his ear could distinguish many more than 12 notes. Stopping off and playing pitches on the paper-covered fingerboard of his original cello in 1924, he would have clearly heard the difference between the chromatic and diatonic semitones, 25/24 and 16/15. He would have heard the difference between the 7/6 and the 6/5, and the difference between the 16/9 and the 15/8.
This must have prompted Partch to push beyond 12 just notes. How many?
The only firm record of a scale between 1923 and 1933 is Partch's own handwritten pencil note on page 37 of the Exposition:
There are 29 ratios ratios within the 2/1, corresponding to my 1928 theory of the more essential tones. The other ratios were comparable to those. The indicated ratios are
1/1 33/32 21/20 15/14 12/11 10/9 8/7 7/6 6/5 11/9 5/4 9/7 4/3 11/8 7/5 10/7 16/11 3/2 14/9 8/5 18/11 5/3 12/7 7/4 9/5 11/6 28/15 40/21 64/33
One possible hint about the development of Partch's scale between 1923 and 1928 comes from an examination of photograph 2 of the chart used in an oral "exposition of monophony" in Pasadena and Los Angeles, Feb-June, 1933 (photo pasted in on page 49 of the 1933 Exposition). The ratios are
12/11 10/9 8/7 6/5 5/4 9/7 9/7 11/8 10/7 3/2 11/7 18/11 12/7 16/9 20/11
and the inversions of these intervals in reverse order
11/10 9/8 7/6 11/9 14/11 4/3 7/5 16/11 14/9 8/5 5/3 7/4 9/5 11/6
plus the 1/1, for a total of 29 tones.
The derivation is elegant: fit one set of tones between the other, and you get a complete scale.
Again, this is what you'd get if you were stopping off strings. By playing the smaller length of the string, you get the higher pitch; by playing the larger length on the other side of the stop, you get the lower pitch. (Playing a string stopped off at 1/3 of its length gives 3/1 on the short side, 4/3 on the long.)
Assume Partch starts with 12 tones derived at least partially from Ellis' duodenarium. Assume he began to stop them off on his cello (this is 1924; he's not yet built his adapted viola): he notices that any set of intervals naturally fall into a complementary set of inversions. From an 11 tone set, he moves to a 15 tone scale in order to get an unfamiliar 7/5 interval and distinguish between the large and small whole tones 9/8 and 10/9. Then he moves rapidly to a 29 tone scale by using the inversions of the 14 tones.
The decision to stop at the 11 limit in 1928 was probably not arbitrary. In his cello experiments in 1924 he must have noticed that ratios of 5 and 7 sounded something like the intervals familiar in Western music. However, when Partch went to the next prime--namely, 11--he would instantly have heard that these intervals sounded like nothing in Western music. The 11/8 is out there. It has NO point of contact with ANYTHING in 19th century theory or practice (Scriabin's "mystic chord" notwithstanding).
Thus, 11-limit represents the first logical step outside the 19th century common practice period of music--11 is the first logical place to go if you're climbing the overtone series and want to extend your system beyond the intervals characteristic of western music.
At this point, Partch is following the footsteps of the pre-classical Greeks. Mark off a string-length, sound it: sound the inversion. Compare the intervals to those produced by other string-lengths.
Dividing and then subdividing string-lengths was a procedure long since known to the Greeks. And on pages 17 and 18 of the 1933 Exposition, Partch shows the process of dividing a string. He draws a series of arcs to indicate divisions and successive subdivisions of various string lengths.
By 1928, Partch has arrived at 29 intervals, a substantial increase over the 12 he'd started with in 1923. But why stop there?
By subdividing already subdivided string lengths, Partch can increase the number of just intervals in his scale without limit.
The original 29 pitches supplemented by 26 additional pitches--that is, 13 additional secondary ratios and their inversions--would have produced the 55 described by Bertha Knisely.
What were those 55 pitches?
At this point, the evidence from black light examination of page 36 of the 1933 Exposition comes in. The original whited-out India-inked ratios at the start of the scale were
1/1 56/55 55/54 45/44 33/32 25/24 12/11
while the original whited-out ratios at the end of the scale were:
28/15 40/21 48/25 27/14 88/45
Note that the single square indicating a pitch contains two notes: 56/55 and 55/54. These pitches fall too close to one another to be easily distinguished, so Partch thought of them as the same note.
This being the case, Partch had arrived at the ultima thule of his process of successive subdivision of intervals. At this point, with several intervals which could not be distinguished by ear, he could go no farther and still produce audibly different results--at least, not using the system of subdividing string lengths as the Greeks had done.
"Ontogeny recapitulates phylogeny," as the anatomists say. Partch's later tunings must have contained in themselves traces of the earlier tuning whence they sprang. This being the case, why not examine all the different scales Partch wrote down between 1928 and 1933, and collapse them into a single super-scale...would this not produce a tuning from which, sometime between 1924 and 1927, he drew his original 29 pitches of the 1928 draft, and then reached into again and again, to draw different sets of pitches between 1928 and 1933?
Adding these pitches to those of Partch's other just scales, we come up with a speculative 55-tone tuning somewhere between 1925 and 1927:
1/1 56/55 55/54 49/48 45/44 33/32 25/24 22/21 16/15 11/10 10/9 9/8 8/7 7/6 32/27 6/5 11/9 5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11 22/15 3/2 32/21 14/9 11/7 8/5 27/16 18/11 5/3 22/16 12/7 7/4 16/9 9/5 20/11 11/6 28/15 15/8 40/21 21/11 48/25 64/33 64/55 27/14 88/45 96/49 110/56 108/55
Except for the inversions of the 56/55 and the 55/54 (which are speculative additions), all these pitches are drawn from one or another tuning Partch wrote down in his own hand between May, 1928 and June, 1933. There is no proof whatever that this super-scale is what he arrived at twixt 1924 and 1926: however, the number of tones is 55, and they do derive from known Partch scales.
Whether or not this is the 55-tone scale of which Bertha Knisely spoke in her 20 December 1933 article, no one can say. It is presented here as a speculative possibility.
(A 57 pitch scale is mentioned in several sources, both dubious. One is a 1932 newspaper article, the other a fleeting mention by Bertha Knisely. However, if there really were at one time 57 rather than 55 pitches, these can easily be obtained by adding a 45/32 tritone and its inversion, the 64/45.)
Having subdivided the octave into more than fifty pitches, somewhere between 1924 and 1928, Partch faced a problem: where to stop?
Once you begin to subdivide a string into integer-ratio lengths, there's no obvious place to end the process.
Partch needed to reach beyond the Greeks--he needed a system more self-contained than the successive subdivision of string-lengths.
5. WHEN DID PARTCH DISCOVER THE TONALITY DIAMOND?
Partch himself tells us about his discovery of the tonality diamond. On page 51, he states
"The 1931 draft admitted undertones as the source of intervals. This was the result of hastened thru reading Henry Cowell's "New Musical Resources."" [The words "the result of" were originally typewritten and have been scrawled out with a pencil; the words "hastened thru" are written over the old phrase.]
Clearly by 1930 Partch needed to find a way to acoustically and mathematically limit the process of deriving just ratios. Otherwise, he would have no way of justifying (all puns intended) why his particular scale--of 37, or 39, or 29, or 55 intervals--was preferable to or more logical than any other set of just pitches.
Reading through Cowell's New Musical Resources in 1930, Partch would have been struck by Cowell's suggestion of using undertones. He began to sketch string lengths of 1, 3, 5, 7, 9, and 11 ... and their undertones. He arranged the two sets of undertones and overtones in a matrix: inspiration!
The result was a mathematically-generated array of pitches inherently limited to 29 tones (36 tones with 5 duplicated 1/1's: 3/3, 5/5, 7/7, 9/9 and 11/11, plus 2 duplicated 3/2s).
This is the most likely scenario, given Partch's own description on page 51 of the 1933 Exposition. However, there is another possibility.
Max Meyer's book The Musician's Arithmetic was published in 1929. On page 22, a tonality diamond appears rotated 90 degrees.
Could Partch have seen this book? Could this have influenced him? Was Meyer's text the original source of the Partch Tonality Diamond, rather than Cowell's book?
It's impossible to be sure.
Although Partch's 1949/1974 diagram of the tonality diamond in Genesis of a Music obscures the derivation from odd numbered harmonics and subharmonics, the diagram of page 22 of the 1933 Exposition shows the derivation very clearly.
This supports the theory that Partch got the idea of the Tonality Diamond after reading Cowell.
On the other hand, Meyer's book was published a year earlier than Cowell's. If Partch read them both, he might first have gotten the idea for the tonality diamond from Meyer. It's worth noting that Meyer's book was published as part of a series, not widely available: Partch would have needed access to a large library to have read Meyer's book.
At this remove it is impossible to be sure whether Partch was inspired to create the Tonality Diamond by Meyer, or by Cowell, or by both authors.
In any case, Partch's own description on page 51 of the Exposition makes the essential situation clear. The idea of the Tonality Diamond must have been lurking in the back of Partch's mind--Cowell's and/or Meyer's book didn't lead him to it, but merely served as midwife to an intuition already waiting to be born.
6-9. WHY THE GAP BETWEEN DRAFTS 1 AND 2 of THE EXPOSITION?
This explains the 3-year gap between Partch's initial start on the draft of the Exposition in 1927, and his decision to revise it in 1931. Partch struggled with methods of organizing and deriving his just scale which would make the tuning seem not merely acoustically palatable, but logically inevitable. In 1931 the breakthrough came: and he revised the entire Exposition to include the tonality diamond. This would also explain why the original 90 pages of the first draft collapsed down to 54 pages by the third draft. Gone were pages and pages of elaborate reasoning, gone were a likely plethora of charts of string subdivisions, gone were various elaborate strategems previously needed to explain why this or that interval ought to be included in the scale, and why another interval ought not to be included.
Instead, Partch now simply wrote down the Tonality Diamond--the nucleus of his scale flowed naturally from this inspiration, a necessary and inevitable limit to the number of the "more essential" pitches. Deriving secondary pitches was a simple matter once the nucleus of the scale was fixed.
We can now estimate the general organization of Partch's original Exposition. The first draft presented Partch's 29 intervals, and spent a great deal of verbiage explaining why these just ratios (and not others) were required.
In 1930, Partch revised his manuscript to reflect his experience with Rudolphine Radil and Callista Rogers. Having accompanied singers on the adapted viola using his just intonation system, Partch would have been prompted to redact his scale so as to eliminate the large gaps near the 1/1 and the 2/1--the better to make it singable.
In 1931, the big breakthrough: the Tonality Diamond. Partch radically rewrote the Exposition, throwing out pages and pages of string-length subdivisions and mathematical argument.
He also added "the principles of song expounded herein," doubtless gleaned from his own convictions about the importance of the voice in music and bolstered by his experience with people who had performed in his system.
Finally, in 1933, he added a history of just intonation to lend his musical system the added luster of historical precedent.
CONCLUSION
While some of the theses set forth in this paper derive from a process of induction, along with an examination of the circumstantial evidence, others are reasonably conclusive.
It remains for a future generation of scholars to extend our knowledge of Partch's tuning systems and the thought processes behind it. For the present, with available evidence, it appears unreasonable to extend the deductions beyond those propounded here.
APPENDIX I - A CHRONOLOGY OF PARTCH'S KNOWN SCALES
| YEAR | NUMBER OF PITCHES | SOURCE |
| 1923 | 12 | 1933 Exposition, page 51 |
| 1924-1927 | 29 | Speculative |
| 1924-1927 | 55 | Bertha Kniseley's Dec. 1933 article |
| 1930 | 57 | 1930 Newspaper article--dubious |
| 1927-1928 | 42 | Original ratios whited-out on page 36 of the Exposition |
| 1928 | 28 | 1933 Exposition, page 37 |
| 1931 | 39 | Page 41 of the Exposition |
| 1932 | 39 | Ratio keyboard chart notarized April, 1932 |
| Feb-June 1933 | 43 | Photo of chart used in oral exposition of monophony in Pasadena & Los Angeles, page 49, Exposition of Monophony |
| June 1933 | 37 | Exposition of Monophony, page 19 |
| June, 1933 | 39 | 1933 Exposition, page 39 |
| August 1933 | 43 | Exposition, page 38 |
| October, 1933 | 41 | Score of "Before the cask of Wine" |
| 1949 | 43 | Genesis Of A Music, 1st Edition |
APPENDIX II - A PARTIAL LIST OF PARTCH'S SCALES IN CHRONOLOGICAL ORDER
STRING FAMILY ADAPTATION - LISTED ON PAGE 41, 1933 "EXPOSITION"
39 TONES - 1925-1928(?)
1/1 49/48 33/32 22/21 16/15 12/11 10/9 9/8 7/5 6/6 11/9
5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11 22/15 3/2 32/21
14/9 11/7 8/5 18/11 5/3 12/7 7/4 16/9 9/5 11/6 15/8 21/11 64/33
96/49
PAGE 37, 1933 "EXPOSITION"
29 TONES - 1928
1/1 33/32 21/10 15/14 12/11 10/9 8/7 7/6 6/5 11/9 5/4 9/7 4/3
11/8 7/5 10/7 16/11 3/2 14/9 8/5 18/11 5/3 12/7 7/4 9/5 11/6
28/15 40/21 64/33
MASTER CHART OF PITCHES FROM PAGE 36 OF 1933 "EXPOSITION"
(underlined pitches from back side of the page, viewed through black light; bracketed pitches from the front of the page) New scale has 41 tone (front side), old scale has 42 tones (back side)
41 - 42 TONES, 1932 - 1933
55/54
1/1 56/55 45/44 33/32 25/24 12/11 11/10 10/9 9/8 8/7 7/6 6/5
11/9 5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11 22/15 3/2
32/21 14/9 11/7 8/5 18/11 5/3 12/7 7/4 16/9 9/5 20/11 11/6
28/15 40/21 48/25 27/14 88/45 [15/8 21/11 64/33 96/49]
"KEYBOARD FABRIC" LISTED ON RATIO KEYBOARD LETTER (SAME AS "STRING FAMILY")
39 TONES - AUGUST 4, 1932
1/1 49/48 33/32 22/21 16/15 12/11 10/9 9/8 7/5 6/6 11/9
5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11 22/15 3/2 32/21
14/9 11/7 8/5 18/11 5/3 12/7 7/4 16/9 9/5 11/6 15/8 21/11 64/33
96/49
PHOTO 2 FROM CHARTS USED IN ORAL EXPOSITION OF MONOPHONY IN PASADENA AND LOS ANGELES FEBRUARY - JUNE, 1933, PAGE 49, 1933, "EXPOSITION"
29 TONES - 1933
1/1 12/11 11/10 10/9 9/8 8/7 7/6 6/5 11/9 5/4 14/11 9/7 4/3
11/8 7/5 10/7 16/11 3/2 14/9 11/7 8/5 18/11 5/3 12/7 7/4 16/9
9/5 20/11 11/6
PHOTO 4: PICTURE OF CHART USED IN ORAL EXPOSITION OF MONOHPONY IN PASADENA
AND LOS ANGELES,FEBRUARY - JUNE 1933 (PAGE 49, 1933 "EXPOSITION")
43 TONE - JUNE 1933
1/1 49/48 33/32 22/21 16/15 12/11 11/10 10/9 9/8 8/7 7/6 32/27
6/5 11/9 5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11
22/15 3/2 32/21 14/9 11/7 8/5 18/11 5/3 27/16 12/7 7/4 16/9
9/5 20/11 11/6 15/8 21/11 64/55 96/49
MANUSCRIPT OF "15. BEFORE THE CASK OF WINE" (ALSO PAGE 39, 1933 "EXPOSITION")
41 TONES - AUGUST 7, 1933
21/11 64/33 96/49 1/1 49/48 33/32 22/21 16/15 12/11 11/10 10/9 9/8 8/7
7/6 32/27 6/5 11/9 5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7
16/11 22/15 3/2 32/21 14/9 11/7 8/5 18/11 5/3 27/16 12/7 16/9 9/5
20/11 11/6 15/8
FROM PAGE 38, 1933 "EXPOSITION" - "NOTATION ADOPTED AUGUST, 1933"
43 TONES - AUGUST 1933
21/11 64/33 96/49 1/1 49/48 33/32 22/21 16/15 12/11 11/10 10/9 9/8
8/7 7/6 32/276/5 11/19 5/4 14/11 9/7 21/16 4/3 15/11 11/8
7/5 10/7 16/11 22/15 3/2 32/21 14/9 11/7 8/5 18/11 5/3 22/16
12/7 7/4 16/9 9/5 20/11 11/6 15/8
PAGE 39, 1933 "EXPOSITION" - "41-TONE SCALE"
41 TONES - AUGUST 7, 1933
21/11 64/33 96/49 1/1 49/48 33/32 22/21 16/15 12/11 11/10 10/9 9/8 8/7
7/6 32/27 6/5 11/9 5/4 14/11 9/7 21/16 4/3 15/11 11/8 7/5 10/7
16/11 22/15 3/2 32/21 14/9 11/7 8/5 18/11 5/3 27/16 12/7 16/9 9/5
20/11 11/6 15/8